Question 7.1: A time series of wave heights is sampled at three-hourly int......

We have Δt = 3 hours, N = 292200 (corresponding to 100 years) and m = 2. The return period is

\frac{292200}{2}\times 3\times \frac{1}{24\times 365.25} = 50years.

The last term on the left-hand side converts units of hours to years (24 hours per day and 365.25 days per year).

It follows that there is a finite chance that the design conditions will be exceeded during the life of a scheme. This probability of exceedance is usually referred to as the ‘return period’.

An event with a return period of T years is likely to be exceeded, on average, once in T years.

The return period should not be confused with the design life of a scheme. For example, if the return period of an extreme event is the same as the design life, then there is a ∼63% chance that the extreme event will be exceeded during the period of the design life. Considering annual maxima, the probability of exceedance and the return period are related by

where L is the design life. Figure 7.3 plots return period against duration for fixed values of probability of exceedance. A related concept that is sometimes used is the degree of security, defined as the probability of a given condition not being exceeded over a fixed period. (This is of course just 1 minus the probability of the condition being exceeded). Graphs of the return period against the degree of security for fixed periods of 2–100 years are drawn in Figure 7.4. So, for example, a structure designed to withstand the 1-in-100-year conditions provides a degree of security of ∼ 0.73 over a period of 30 years.

Figure 7.5 plots the return period against duration for fixed levels of the degree of security. This type of plot is useful when an engineer has to prepare a design on the basis of a specified probability of failure over the (known) design life. The required return period of an extreme event used for design can be read from the graph. For example, if a degree of security of 0.9 is required for a structure with a design life of 10 years, then the structure must be designed against 1-in-100-year conditions.

In some cases only the maximum value over a given period is recorded. For example, often only the annual maximum water level is recorded. We have, therefore, one event per year, and this event could have occurred at any time throughout the year. To answer the last question, (iii), the N-year return value, q_{N}, is calculated from the equation

F(qn)= \begin{cases} 1-\frac{1}{N}     for  maxima \\ \frac{1}{N}        for  minima \end{cases}                  (7.4)

where F(q) is the cumulative distribution function of the annual maximum overtopping values.